DOCSLIB.ORG

Manifold Interpolation and Model Reduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Manifold Interpolation and Model Reduction MANIFOLD INTERPOLATION AND MODEL REDUCTION RALF ZIMMERMANN∗ Abstract. One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction, (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. H. A. Schilders, L. M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of model reduction. Moreover, the principal approaches to data interpolation and Taylor-like extrapolation on matrix manifolds are outlined and complemented by algorithms in pseudo-code. Key words. parametric model reduction, matrix manifold, Riemannian computing, geodesic interpolation, interpolation on manifolds, Grassmann manifold, Stiefel manifold, matrix Lie group AMS subject classifications. 15-01, 15A16, 15B10, 15B48, 53-04, 65F60, 41-01, 41A05, 65F99, 93A15, 93C30 1. Introduction & Motivation. This work addresses interpolation approaches for parametric model reduction. This includes techniques for • computing trajectories of parameterized subspaces, • computing trajectories of parameterized reduced orthogonal bases, • structure-preserving interpolation. Mathematically, this requires data processing on nonlinear matrix manifolds. The exposition at hand intends to be an introduction and a reference guide to numerical procedures with matrix manifold-valued data. As such it addresses practitioners and scientists new to the field. It covers the essentials of those matrix manifolds that arise most frequently in practical problems in model reduction. The main purpose is not to discuss concrete model reduction applications, but rather to provide the essential tools, building blocks and background theory to enable the reader to devise her/his own approaches for such applications. The text was designed such that it works as a commented formula collection, meanwhile giving sufficient context, explanations and, not least, precise references to enable the interested reader to immerse further in the topic. 1.1. Parametric model reduction via manifold interpolation: An intro- ductory example. The basic objective in model reduction is to emulate a large-scale dynamical system with very few degrees of freedom such that its input/output behav- ior is preserved as well as possible. While classical model reduction techniques aim at producing an accurate low-order approximation to the autonomous behavior of the arXiv:1902.06502v2 [math.NA] 11 Sep 2019 original system, parametric model reduction (pMOR) tries to account for additional system parameters. If we look for instance at aircraft aerodynamics, an important task is to solve the unsteady Navier-Stokes equations at various flight conditions, which are, amongst others, specified by the altitude, the viscosity of the fluid (i.e. the Reynolds number) and the relative velocity (i.e. the Mach number).We explain the objective of pMOR with the aid of a generic example in the context of proper orthogo- nal decomposition-based model reduction. Similar considerations apply to frequency domain approaches, Krylov subspace methods and balanced truncation, which are ∗Department of Mathematics and Computer Science, University of Southern Denmark (SDU) Odense, ([email protected]). 1 discussed in other chapters of the upcoming Handbook on Model Order Reduction. Consider a spatio-temporal dynamical system in semi-discrete form @ x(t; µ) = f(x(t; µ); µ); x(t ; µ) = x ; (1.1) @t 0 0,µ where x(t; µ) 2 Rn is the spatially discretized state vector of dimension n, the vec- d tor µ = (µ1; : : : ; µd) 2 R accounts for additional system parameters and f( · ; µ): Rn ! Rn is the (possibly nonlinear, parameter-dependent) right hand side function. Projection-based MOR starts with constructing a suitable low-dimensional subspace that acts as a space of candidate solutions. Subspace construction. One way to construct the required projection subspace is the proper orthogonal decomposition (POD), [48].In its simplest form, the POD 1 can be summarized as follows. For a fixed system parameter µ = µ0, let x := m n x(t1; µ0); :::; x := x(tm; µ0) 2 R be a set of state vectors satisfying (1.1) and let 1 m n×m i S := x ; :::; x 2 R . The state vectors x are called snapshots and the matrix S is called the associated snapshot matrix. POD is concerned with finding a subspace n×r V of dimension r ≤ m represented by a column-orthogonal matrix Vr 2 R such that the error between the input snapshots and their orthogonal projection onto V = ran(Vr) is minimized: X k T k 2 T 2 min kx − VV x k2 , min kS − VV SkF : V 2 n×r ;V T V =I V 2 n×r ;V T V =I R k R The main result of POD is that for any r ≤ m, the best r-dimensional approximation of ran(x1; :::; xm) in the above sense is V = ran(v1; :::; vr), where fv1; :::; vrg are the eigenvectors of the matrix SST corresponding to the r largest eigenvalues. The sub- 1 r space V is called the POD subspace and the matrix Vr = (v ; :::; v ) is the POD basis matrix. The same subspace is obtained via a compact singular value decomposition (SVD) of the snapshot matrix S = VΣZT , truncated to the first r ≤ m columns of n×m V 2 R by setting V := ran(Vr). For more details, see, e.g. [17, x3.3]. In the following, we drop the index r and assume that V is already the truncated matrix V = (v1; :::; vr) 2 Rn×r. Since the input snapshots are supplied at a fixed system parameter vector µ0, the POD subspace is considered to be an appropriate space of solution candidates V(µ0) = ran(V(µ0)) at µ0. Projection. POD leads to a parameter decoupling x~(t; µ0) = V(µ0)xr(t): (1.2) In this way, the time trajectory of the reduced model is uniquely defined by the coef- r ficient vector xr(t) 2 R that represents the reduced state vector with respect to the subspace ran(V(µ0)). Given a matrix W(µ0) such that the matrix pair V(µ0); W(µ0) T is bi-orthogonal, i.e. W(µ0) V(µ0) = I, the original system (1.1) can be reduced in T dimension as follows. Substituting (1.2) in (1.1) and multiplying with W(µ0) from the left leads to d x (t) = T (µ )f( (µ )x (t); µ ); x (t ) = T (µ )x : (1.3) dt r W 0 V 0 r 0 r 0 V 0 0,µ0 This approach goes by the name of Petrov-Galerkin projection, if W(µ0) 6= V(µ0) and Galerkin projection if W(µ0) = V(µ0). There are various ways to proceed from (1.3) 2 depending on the nature of the function f and many of them are discussed in other chapters of the upcoming Handbook on Model Order Reduction. 1 For illustration purposes, we proceed with W(µ0) = V(µ0) and assume that the right hand side function f splits into a linear and a nonlinear part: f(x; µ0) = n×n A(µ0)x + f(x; µ0), where A(µ0) 2 R is, say, a symmetric and negative definite matrix to foster stability. Then, (1.3) becomes d x (t) = T (µ )A(µ ) (µ )x (t) + T (µ )f (µ )x (t); µ : dt r V 0 0 V 0 r V 0 V 0 r 0 In the discrete empirical interpolation method (DEIM, [27]), the large-scale nonlinear n×s term f V(µ0)xr(t); µ0) is approximated via a mask matrix P = (ei1 ; : : : ; eis ) 2 R , j T n where fi1; : : : ; isg ⊂ f1; : : : ; ng and ej = (:::; 1;:::) 2 R is the jth canonical unit vector. The mask matrix P acts as an entry selector on a given n-vector via T T s n×s P v = (vi1 ; : : : ; vis ) 2 R . In addition, another POD basis matrix U(µ0) 2 R is used, which is obtained from snapshots of the nonlinear term. The matrices P and U(µ0) are combined to form an oblique projection of the non-linear term onto the subspace ran(U(µ0)). This leads to the reduced model d x (t) = T (µ )A(µ ) (µ )x (t) dt r V 0 0 V 0 r T T −1 T +V (µ0)U(µ0)(P U(µ0)) P f V(µ0)xr(t); µ0 ; (1.4) whose computational complexity is formally independent of the full-order dimension T n, see [27] for details. Mind that by assumption, M(µ0) := −V (µ0)A(µ0)V(µ0) is symmetric positive definite and that both V(µ0) and U(µ0) are column-orthogonal. Moreover, for a fixed mask matrix P , coordinate changes of V(µ0) and U(µ0) do not affect the approximated statex ~(t; µ0) = V(µ0)xr(t), so that essentially, the reduced system (1.4) depends only on the subspaces ran(V(µ0)) and ran(U(µ0)) rather than 2 the matrices V(µ0) and U(µ0). Solving (1.3), (1.4) constitutes the online stage of model reduction. The main focus of this exposition is not on the efficient solution of the reduced systems (1.3) or (1.4) at a fixed µ0, but on tackling parametric variations in µ. In view of the associated computational costs, it is important that this can be achieved without computing additional snapshots in the online stage. A straightforward way to achieve this is to extend the snapshot sampling to the µ- parameter range to produce POD basis matrices that are to cover all input parameters.
Load more

Recommended publications
  • The Grassmann Manifold
    The Grassmann Manifold 1. For vector spaces V and W denote by L(V; W ) the vector space of linear maps from V to W . Thus L(Rk; Rn) may be identified with the space Rk£n of k £ n matrices. An injective linear map u : Rk ! V is called a k-frame in V . The set k n GFk;n = fu 2 L(R ; R ): rank(u) = kg of k-frames in Rn is called the Stiefel manifold. Note that the special case k = n is the general linear group: k k GLk = fa 2 L(R ; R ) : det(a) 6= 0g: The set of all k-dimensional (vector) subspaces ¸ ½ Rn is called the Grassmann n manifold of k-planes in R and denoted by GRk;n or sometimes GRk;n(R) or n GRk(R ). Let k ¼ : GFk;n ! GRk;n; ¼(u) = u(R ) denote the map which assigns to each k-frame u the subspace u(Rk) it spans. ¡1 For ¸ 2 GRk;n the fiber (preimage) ¼ (¸) consists of those k-frames which form a basis for the subspace ¸, i.e. for any u 2 ¼¡1(¸) we have ¡1 ¼ (¸) = fu ± a : a 2 GLkg: Hence we can (and will) view GRk;n as the orbit space of the group action GFk;n £ GLk ! GFk;n :(u; a) 7! u ± a: The exercises below will prove the following n£k Theorem 2. The Stiefel manifold GFk;n is an open subset of the set R of all n £ k matrices. There is a unique differentiable structure on the Grassmann manifold GRk;n such that the map ¼ is a submersion.
    [Show full text]
  • Cheap Orthogonal Constraints in Neural Networks: a Simple Parametrization of the Orthogonal and Unitary Group
    Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group Mario Lezcano-Casado 1 David Mart´ınez-Rubio 2 Abstract Recurrent Neural Networks (RNNs). In RNNs the eigenval- ues of the gradient of the recurrent kernel explode or vanish We introduce a novel approach to perform first- exponentially fast with the number of time-steps whenever order optimization with orthogonal and uni- the recurrent kernel does not have unitary eigenvalues (Ar- tary constraints. This approach is based on a jovsky et al., 2016). This behavior is the same as the one parametrization stemming from Lie group theory encountered when computing the powers of a matrix, and through the exponential map. The parametrization results in very slow convergence (vanishing gradient) or a transforms the constrained optimization problem lack of convergence (exploding gradient). into an unconstrained one over a Euclidean space, for which common first-order optimization meth- In the seminal paper (Arjovsky et al., 2016), they note that ods can be used. The theoretical results presented unitary matrices have properties that would solve the ex- are general enough to cover the special orthogo- ploding and vanishing gradient problems. These matrices nal group, the unitary group and, in general, any form a group called the unitary group and they have been connected compact Lie group. We discuss how studied extensively in the fields of Lie group theory and Rie- this and other parametrizations can be computed mannian geometry. Optimization methods over the unitary efficiently through an implementation trick, mak- and orthogonal group have found rather fruitful applications ing numerically complex parametrizations usable in RNNs in recent years (cf.
    [Show full text]
  • A Geometric Take on Metric Learning
    A Geometric take on Metric Learning Søren Hauberg Oren Freifeld Michael J. Black MPI for Intelligent Systems Brown University MPI for Intelligent Systems Tubingen,¨ Germany Providence, US Tubingen,¨ Germany [email protected] [email protected] [email protected] Abstract Multi-metric learning techniques learn local metric tensors in different parts of a feature space. With such an approach, even simple classifiers can be competitive with the state-of-the-art because the distance measure locally adapts to the struc- ture of the data. The learned distance measure is, however, non-metric, which has prevented multi-metric learning from generalizing to tasks such as dimensional- ity reduction and regression in a principled way. We prove that, with appropriate changes, multi-metric learning corresponds to learning the structure of a Rieman- nian manifold. We then show that this structure gives us a principled way to perform dimensionality reduction and regression according to the learned metrics. Algorithmically, we provide the first practical algorithm for computing geodesics according to the learned metrics, as well as algorithms for computing exponential and logarithmic maps on the Riemannian manifold. Together, these tools let many Euclidean algorithms take advantage of multi-metric learning. We illustrate the approach on regression and dimensionality reduction tasks that involve predicting measurements of the human body from shape data. 1 Learning and Computing Distances Statistics relies on measuring distances. When the Euclidean metric is insufficient, as is the case in many real problems, standard methods break down. This is a key motivation behind metric learning, which strives to learn good distance measures from data.
    [Show full text]
  • Building Deep Networks on Grassmann Manifolds
    Building Deep Networks on Grassmann Manifolds Zhiwu Huangy, Jiqing Wuy, Luc Van Goolyz yComputer Vision Lab, ETH Zurich, Switzerland zVISICS, KU Leuven, Belgium fzhiwu.huang, jiqing.wu, [email protected] Abstract The popular applications of Grassmannian data motivate Learning representations on Grassmann manifolds is popular us to build a deep neural network architecture for Grassman- in quite a few visual recognition tasks. In order to enable deep nian representation learning. To this end, the new network learning on Grassmann manifolds, this paper proposes a deep architecture is designed to accept Grassmannian data di- network architecture by generalizing the Euclidean network rectly as input, and learns new favorable Grassmannian rep- paradigm to Grassmann manifolds. In particular, we design resentations that are able to improve the final visual recog- full rank mapping layers to transform input Grassmannian nition tasks. In other words, the new network aims to deeply data to more desirable ones, exploit re-orthonormalization learn Grassmannian features on their underlying Rieman- layers to normalize the resulting matrices, study projection nian manifolds in an end-to-end learning architecture. In pooling layers to reduce the model complexity in the Grass- summary, two main contributions are made by this paper: mannian context, and devise projection mapping layers to re- spect Grassmannian geometry and meanwhile achieve Eu- • We explore a novel deep network architecture in the con- clidean forms for regular output layers. To train the Grass- text of Grassmann manifolds, where it has not been possi- mann networks, we exploit a stochastic gradient descent set- ble to apply deep neural networks.
    [Show full text]
  • On Manifolds of Tensors of Fixed Tt-Rank
    ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest.
    [Show full text]
  • INTRODUCTION to ALGEBRAIC GEOMETRY 1. Preliminary Of
    INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1. Preliminary of Calculus on Manifolds 1.1. Tangent Vectors. What are tangent vectors we encounter in Calculus? 2 0 (1) Given a parametrised curve α(t) = x(t); y(t) in R , α (t) = x0(t); y0(t) is a tangent vector of the curve. (2) Given a surface given by a parameterisation x(u; v) = x(u; v); y(u; v); z(u; v); @x @x n = × is a normal vector of the surface. Any vector @u @v perpendicular to n is a tangent vector of the surface at the corresponding point. (3) Let v = (a; b; c) be a unit tangent vector of R3 at a point p 2 R3, f(x; y; z) be a differentiable function in an open neighbourhood of p, we can have the directional derivative of f in the direction v: @f @f @f D f = a (p) + b (p) + c (p): (1.1) v @x @y @z In fact, given any tangent vector v = (a; b; c), not necessarily a unit vector, we still can define an operator on the set of functions which are differentiable in open neighbourhood of p as in (1.1) Thus we can take the viewpoint that each tangent vector of R3 at p is an operator on the set of differential functions at p, i.e. @ @ @ v = (a; b; v) ! a + b + c j ; @x @y @z p or simply @ @ @ v = (a; b; c) ! a + b + c (1.2) @x @y @z 3 with the evaluation at p understood.
    [Show full text]
  • Lecture 2 Tangent Space, Differential Forms, Riemannian Manifolds
    Lecture 2 tangent space, differential forms, Riemannian manifolds differentiable manifolds A manifold is a set that locally look like Rn. For example, a two-dimensional sphere S2 can be covered by two subspaces, one can be the northen hemisphere extended slightly below the equator and another can be the southern hemisphere extended slightly above the equator. Each patch can be mapped smoothly into an open set of R2. In general, a manifold M consists of a family of open sets Ui which covers M, i.e. iUi = M, n ∪ and, for each Ui, there is a continuous invertible map ϕi : Ui R . To be precise, to define → what we mean by a continuous map, we has to define M as a topological space first. This requires a certain set of properties for open sets of M. We will discuss this in a couple of weeks. For now, we assume we know what continuous maps mean for M. If you need to know now, look at one of the standard textbooks (e.g., Nakahara). Each (Ui, ϕi) is called a coordinate chart. Their collection (Ui, ϕi) is called an atlas. { } The map has to be one-to-one, so that there is an inverse map from the image ϕi(Ui) to −1 Ui. If Ui and Uj intersects, we can define a map ϕi ϕj from ϕj(Ui Uj)) to ϕi(Ui Uj). ◦ n ∩ ∩ Since ϕj(Ui Uj)) to ϕi(Ui Uj) are both subspaces of R , we express the map in terms of n ∩ ∩ functions and ask if they are differentiable.
    [Show full text]
  • DIFFERENTIAL GEOMETRY COURSE NOTES 1.1. Review of Topology. Definition 1.1. a Topological Space Is a Pair (X,T ) Consisting of A
    DIFFERENTIAL GEOMETRY COURSE NOTES KO HONDA 1. REVIEW OF TOPOLOGY AND LINEAR ALGEBRA 1.1. Review of topology. Definition 1.1. A topological space is a pair (X; T ) consisting of a set X and a collection T = fUαg of subsets of X, satisfying the following: (1) ;;X 2 T , (2) if Uα;Uβ 2 T , then Uα \ Uβ 2 T , (3) if Uα 2 T for all α 2 I, then [α2I Uα 2 T . (Here I is an indexing set, and is not necessarily finite.) T is called a topology for X and Uα 2 T is called an open set of X. n Example 1: R = R × R × · · · × R (n times) = f(x1; : : : ; xn) j xi 2 R; i = 1; : : : ; ng, called real n-dimensional space. How to define a topology T on Rn? We would at least like to include open balls of radius r about y 2 Rn: n Br(y) = fx 2 R j jx − yj < rg; where p 2 2 jx − yj = (x1 − y1) + ··· + (xn − yn) : n n Question: Is T0 = fBr(y) j y 2 R ; r 2 (0; 1)g a valid topology for R ? n No, so you must add more open sets to T0 to get a valid topology for R . T = fU j 8y 2 U; 9Br(y) ⊂ Ug: Example 2A: S1 = f(x; y) 2 R2 j x2 + y2 = 1g. A reasonable topology on S1 is the topology induced by the inclusion S1 ⊂ R2. Definition 1.2. Let (X; T ) be a topological space and let f : Y ! X.
    [Show full text]
  • Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes Jean-Daniel Boissonnat, Leonidas J
    Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Oudot To cite this version: Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Oudot. Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes. Discrete and Computational Geometry, Springer Verlag, 2009, pp.37. hal-00488434 HAL Id: hal-00488434 https://hal.archives-ouvertes.fr/hal-00488434 Submitted on 2 Jun 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes Jean-Daniel Boissonnat Leonidas J. Guibas Steve Y. Oudot INRIA, G´eom´etrica Team Dept. Computer Science Dept. Computer Science 2004 route des lucioles Stanford University Stanford University 06902 Sophia-Antipolis, France Stanford, CA 94305 Stanford, CA 94305 [email protected] [email protected] [email protected]∗ Abstract It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density.
    [Show full text]
  • 5. the Inverse Function Theorem We Now Want to Aim for a Version of the Inverse Function Theorem
    5. The inverse function theorem We now want to aim for a version of the Inverse function Theorem. In differential geometry, the inverse function theorem states that if a function is an isomorphism on tangent spaces, then it is locally an isomorphism. Unfortunately this is too much to expect in algebraic geometry, since the Zariski topology is too weak for this to be true. For example consider a curve which double covers another curve. At any point where there are two points in the fibre, the map on tangent spaces is an isomorphism. But there is no Zariski neighbourhood of any point where the map is an isomorphism. Thus a minimal requirement is that the morphism is a bijection. Note that this is not enough in general for a morphism between al- gebraic varieties to be an isomorphism. For example in characteristic p, Frobenius is nowhere smooth and even in characteristic zero, the parametrisation of the cuspidal cubic is a bijection but not an isomor- phism. Lemma 5.1. If f : X −! Y is a projective morphism with finite fibres, then f is finite. Proof. Since the result is local on the base, we may assume that Y is affine. By assumption X ⊂ Y × Pn and we are projecting onto the first factor. Possibly passing to a smaller open subset of Y , we may assume that there is a point p 2 Pn such that X does not intersect Y × fpg. As the blow up of Pn at p, fibres over Pn−1 with fibres isomorphic to P1, and the composition of finite morphisms is finite, we may assume that n = 1, by induction on n.
    [Show full text]
  • Stiefel Manifolds and Polygons
    Stiefel Manifolds and Polygons Clayton Shonkwiler Department of Mathematics, Colorado State University; [email protected] Abstract Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons – or even to formalize what a random polygon is – it is convenient to interpret each polygon as a point in some parameter space, essentially trading the complexity of the object for the complexity of the space. In this paper I describe such an interpretation where the parameter space is an abstract but very nice space called a Stiefel manifold and show how to exploit the geometry of the Stiefel manifold both to generate random polygons and to morph one polygon into another. Introduction The goal is to describe an identification of polygons with points in certain nice parameter spaces. For example, every triangle in the plane can be identified with a pair of orthonormal vectors in 3-dimensional 3 3 space R , and hence planar triangle shapes correspond to points in the space St2(R ) of all such pairs, which is an example of a Stiefel manifold. While this space is defined somewhat abstractly and is hard to visualize – for example, it cannot be embedded in Euclidean space of fewer than 5 dimensions [9] – it is easy to visualize and manipulate points in it: they’re just pairs of perpendicular unit vectors. Moreover, this is a familiar and well-understood space in the setting of differential geometry. For example, the group SO(3) of rotations of 3-space acts transitively on it, meaning that there is a rotation of space which transforms any triangle into any other triangle.
    [Show full text]
  • Hodge Theory
    HODGE THEORY PETER S. PARK Abstract. This exposition of Hodge theory is a slightly retooled version of the author's Harvard minor thesis, advised by Professor Joe Harris. Contents 1. Introduction 1 2. Hodge Theory of Compact Oriented Riemannian Manifolds 2 2.1. Hodge star operator 2 2.2. The main theorem 3 2.3. Sobolev spaces 5 2.4. Elliptic theory 11 2.5. Proof of the main theorem 14 3. Hodge Theory of Compact K¨ahlerManifolds 17 3.1. Differential operators on complex manifolds 17 3.2. Differential operators on K¨ahlermanifolds 20 3.3. Bott{Chern cohomology and the @@-Lemma 25 3.4. Lefschetz decomposition and the Hodge index theorem 26 Acknowledgments 30 References 30 1. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory. In Section 2, we first describe a key motivation behind the Hodge theory for compact, closed, oriented Riemannian manifolds: the observation that the differential forms that satisfy certain par- tial differential equations depending on the choice of Riemannian metric (forms in the kernel of the associated Laplacian operator, or harmonic forms) turn out to be precisely the norm-minimizing representatives of the de Rham cohomology classes. This naturally leads to the statement our first main theorem, the Hodge decomposition|for a given compact, closed, oriented Riemannian manifold|of the space of smooth k-forms into the image of the Laplacian and its kernel, the sub- space of harmonic forms. We then develop the analytic machinery|specifically, Sobolev spaces and the theory of elliptic differential operators|that we use to prove the aforementioned decom- position, which immediately yields as a corollary the phenomenon of Poincar´eduality.
    [Show full text]